1. Field
The present disclosure relates to the field of mechanical kinematic mounts.
2. Related Art
A kinematic mount (or coupling) is a mechanism that connects two subassemblies (referred to in this disclosure as subassemblies A and B) and constrains exactly all of the possible degrees of freedom (DOFs) between them. For any two free rigid subassemblies in three dimensions, there are six such DOFs—three translational and three rotational. When subassemblies are kinematically coupled, the coupling is stress-free and repeatable. The subassemblies can be complex structures, or simple rigid bodies.
A kinematic mount is comprised of “mating elements” that connect the two subassemblies. Each mating element is comprised of two mating “components”—one attached to each subassembly. The connection of the two subassemblies occurs when they are brought toward each other (with their pre-attached components of the mating elements) along a “mating direction” or “mating axis” until each pair of components connects to form its complete mating element.
Generally, the components of a mating element do not connect in a unique unambiguous way—they have “wiggle room” within them. It is the property of the entire kinematic mount (indeed, it is its underlying quality) that when all mating elements are engaged, it collectively eliminates all of the “Wiggle” in all of the mating elements, and does so in such a way as to not over-eliminate, or over-constrain then, so they are not “fighting each other”.
The wiggle comes into play, for example, when one of the subassemblies warps or expands due to a change in temperature. Inside the mating elements, the components may shift within their wiggle room, but overall the kinematic mount will remain well-constrained, and no stresses will develop.
Typically the mating elements are located roughly on a plane that is perpendicular to the mating direction—the “Mating Plane”.
Kinematic mounts are a special case of kinematic structures, which also constrain exactly six DOFs, but do not have the ability to come apart neatly and re-connect along a single direction, and so have to be assembled in place.
There are several reasons why Kinematic mounts are used.
If one subassembly is the foundation for an optical instrument, and the other subassembly is the optical instrument itself, then it is guaranteed that the instrument won't be warped by the foundation, even if the foundation (or instrument) warps as a result of temperature change, ground shift, or mechanical stress due to reason such as tilting of the foundation (as would be the case in a telescope that tracks stars across the sky)
Additionally, every time the mount is taken apart and put together, the subassemblies retain the same spatial relationship with respect to each other. In the case of the telescope, if the operator has several observation cameras that need to be used interchangeably, mounting them on a kinematic mount guarantees that on each use they'll return to the position they were at when last disassembled. Similarly, a robot that connects to a production tool using a kinematic mount is guaranteed that the tool is held in exactly the same way every time, and so does not have to be re-calibrated.
An important property of a kinematic mount is its load carrying capacity. In many implementations of kinematic mounts, because their structure relies on point contacts between spheres and flat surfaces, kinematic mounts are limited to very light loads, and are damaged if this load is exceeded because the spheres dent the flat surfaces. This is clearly a problem when a telescope camera weighs more than a ton, or the robot has to lift a roof of a car quickly into position.
Another important property is low friction. If a sphere presses into a flat surface, even before it permanently dents it and only elastically deforms it, it becomes difficult for the sphere to slide, since it has to “drag the elastic dent” with it.
Kinematic couplings are old art. For example FIGS. 1A and 1B illustrate a “cone-groove-flat” mount, subassembly A [10] has three spherical protrusions [11], and subassembly B [12] has a conical depression [13], a V-groove depression [14], and a flat area [15]. If the subassemblies are mated such that the spherical protrusions of subassembly A are tangent to the mating surfaces of the components of subassembly B, then the spatial relation between subassemblies A and B is unique —the conical depression eliminates three DOFs, the V-groove eliminates two DOFs, and the flat area eliminates the last of the six DOFs. These mating elements are each an example, respectively, of a 3-DOF, a 2-DOF, and a 1-DOF mating element.
A similar scheme is the “three-groove” mount (FIGS. 2A and 2B), in which three spherical protrusions in subassembly A [20] fit into three V-groove depressions in subassembly B [21], such that the spherical protrusions are tangent to the faces of the V grooves, eliminating two DOFs each. This mount has an advantage over the “cone-groove-flat” mount in that its three mating elements are identical, and so the mount is symmetrical. The three-groove mount is also a widely used mechanism.
In the two examples given above, the kinematic mounts are comprised of three mating elements. The components of the mating elements can be embodied as features that are formed directly into the subassembly (such as the groove and conical depressions in these examples) or as bodies that are mechanically attached to them (such as the spherical protrusions in these examples).
Another property of kinematic mounts is that the precise location of the components is not critical. As long as all pairs of components can contact each other simultaneously and form their respective mating elements, the mount will function and the assembly will be kinematic.
For example, in the case of the cone-groove-flat mount, it does not matter where a spherical protrusion fits along the axis of its matching V-groove, or within its matching flat area. This is the “Wiggle room” referred to earlier.
Similarly, in the case of the three-groove mount, the axes of the v-grooves do not need to precisely intersect at a single point, as long as they are pointing roughly towards a common center. A deviation of up to 15° from the centroid of the mating elements hardly detracts from the functionality of the mount. Only if one of the v-grooves is fabricated so far out of alignment that it points at (or nearly at) one of the other v-grooves does the mount fail to function properly.
This means that there is no need to require tight manufacturing tolerances to achieve the stress-free, and repeatable coupling. The tolerances on the shape of the individual components (spheres, cone, V-groove, flat) are assumed to be much tighter than the positional tolerances of the mating elements and so are assumed to be perfect. For example, the cone is assumed to touch the sphere along a circle (whereas an imperfect cone would touch the sphere in three points and might even be able to rock in it.) This is a practical and realistic assumption, since it is relatively easy to achieve these tolerances.
When designing the mating elements, it is important to consider that as the kinematic mount is assembled by moving its subassemblies toward each other in the Mating Direction, the components of the mating elements must be able to come together, in that direction, without interference.
The mating elements are only typically expected to work in compression, as if the bottom subassembly is attached to the floor, and the only force acting on the top subassembly is gravity. In this disclosure, when referring to a “top” and “bottom” subassembly, such an orientation is assumed.
However, in some situations, such as on a telescope that tilts from vertical to horizontal, an airplane that can fly upside-down, or on a vehicle that is bouncing on a rocky road, the mating elements might experience tensile forces, trying to pull them apart, and clearly the mating elements shown so far are unable to counteract such forces. (as are any mating elements that are based on a “seating” of one body within another)
In such cases either an external load is placed on the subassemblies to ensure that they are always loaded in compression, or a “retainer” mechanism is added to the mating element, which pulls its two components together to overcome any force that is trying to pull them apart. Such a mechanism must be compliant so as not interfere with the positioning functionality of the mating element.
For example, if the two components of the mating element were magnetic, or if magnets were embedded in the subassemblies right near the mating element locations, such that the magnets almost touch each other when the mating elements are assembled, then that would constitute a retainer. This practice is sometimes used, but usually magnetic retainers are not considered secure enough since a sharp jolt can disengage them, and mechanical retainers are used instead.
The design of a kinematic mount can often be divided into its abstract geometry (How many and which DOFs are eliminated at each mating element) and into the embodiments or implementations of its mating elements (Is the 2-DOF mating element implemented as a sphere-in-groove or is some other mechanism used?).
Of the first aspect, a common property of existing mounts is that they each rely on three mating elements. The underlying assumption is that a four-legged kinematic mount is not feasible for the same reason that a four-legged restaurant table rocks on an uneven floor, whereas a three-legged table is always stable. This assumption, however, is only relevant to kinematic mounts if they decouple any in-plane motion from out-of-plane motion, as existing art mounts do. (For example, the motion of the spherical protrusion inside the V-groove is purely horizontal, and does not result in vertical motion)
A different way to phrase this is that the restaurant table is assumed to be free to move in the plane of the floor (X-Y-Theta) and so only three DOFs need to be constrained (tip-tilt-Z). Because of this, after three legs impose one DOF constraint each, any additional legs result in an over-constrained system which results in rocking since unless one leg is in the air, the system has no solution. However, a 3D spatial assembly has six DOFs to constrain, and so there is no theoretical impediment to creating a mount using more than three mating elements.
It is in fact possible to create a four-mating-element kinematic mount by combining in-plane and out-of-plane motion—which is something a typical restaurant table does not do. This coupling of in-plane and out-of-plane is very important, and explained further below.
The utility of a four-legged kinematic mount is that most real life subassemblies—such as optical plates, boxes, cars, and washing machines—are rectangular in shape and so are naturally assembled or held using four mating elements rather than three.
In this disclosure, the term “Pivot” refers to a mating element that constrains exactly three translational DOFs, allowing only rotational DOFs. The Pivot constrains two points, each fixed in relation to one of its components, to remain coincident.
In this disclosure, the term “Slider” refers to a mating element that constrains exactly two translational DOFs, allowing rotational DOFs and one translational DOF. The Slider allows a point that is fixed in relation to one of its components to move along a one-dimensional curve that is fixed in relation to the other components.
In this disclosure, the term “Spacer” refers to a mating elements that constrain exactly one translational DOF, allowing rotational DOFs and two translational DOFs. The Space maintains a fixed distance between two points, each of them fixed in relation to one of the components of the Spacer.
Of the second aspect, implementing the mating elements in such a way that they function well in practical situations can be complicated, especially in regards to load carrying capacity and friction.
In this disclosure, the term “Conoid” refers to a body with an internal concave surface of revolution (or a portion thereof) that is generated by revolving a straight line or a concave curve, in such a way that if a matching solid sphere is pushed into it, the contact area will be a circle or a portion thereof, perpendicular to the axis of revolution, and centered on it, and the Conoid surface will be tangent to the sphere.
In this disclosure, the term “Spheroid” refers to a body with an external convex surface that is a sphere or a portion thereof a sphere.
In this disclosure, the term “CS Pair” or “CS interface” refers to a matching Conoid-Spheroid pair. The main property of a CS pair is that the Conoid can rotate freely around the center of the Spheroid in all three axes. (Or alternatively, the Spheroid can rotate within the Conoid)
CS Pairs are great building blocks for mating elements since they have a high load carrying capacity, and the rotation of the Spheroid within the Conoid has very low friction. Also, since both parts are bodies of revolution, they can be manufactured easily and with high fidelity using a lathe.
It is best to fabricate a CS Pair such that the Spheroid is both stronger and more rigid than the Conoid. This cause any deformation (whether elastic or plastic) to occur in the Conoid, while the Spheroid remains undeformed and spherical, reducing any impediment to motion.
The easiest type of mating element to implement is the Pivot. FIG. 3 shows one implementation, using a single CS Pair. A Spheroid [30] is attached to subassembly A [32], and a Conoid [31] is attached to subassembly B [33]. The interface between them allows subassembly B to rotate around the center of the spherical surface, but does allow it to move.
A Pivot can also be implemented (for example) using a plain spherical bearing or other common mechanisms.
The implementations of Slider mating elements are more problematic. A common problem with them is that they often involve mechanical single point contacts, as in the contact between a spherical protrusion and a V-groove or a flat. Point contacts are limited in the amount of load they can hold. Beyond a certain limit, the stress concentration permanently deforms the mating surfaces, and the deformed parts no longer function properly.
In U.S. Pat. Nos. 6,729,589 and 7,173,779 is described a Slider mating element comprised of spherical and cylindrical surfaces (A “Spherolinder”) that interface a V-groove and a conical depression without creating point contacts, only line contacts (FIG. 4). The Spherolinder mating element behaves very similarly to a sphere-in-groove mating element, but greatly increases the load carrying capacity.
In U.S. patent application Ser. No. 13/032,607 is described a different (“Bead”) mating element that achieves a similar result, but is easier to fabricate since it replaces the V-groove feature with a round shaft (FIG. 5).
Both these inventions create an equivalent mechanism to a sphere sliding inside a V-groove, along a straight line. However, it is not absolutely necessary for the sphere to move along a straight line. It is only important that it travels along a one dimensional construct, and so if the Sphere travels along an arc, for example, the mating element will still be a Slider, and function correctly within the context of a kinematic mount. Such a Slider implementation is described further below.
Using this definition, the three-groove mount of FIGS. 2A and 2B is a special case of a three-Slider mount, as are the three-Spherolinder mount and three-Bead mount referred to above.
In both the disclosures mentioned above, the Spherolinder and Bead mating elements are used to replace ball-in-V-groove Slider elements within a traditional three-Slider kinematic mount, resulting in an otherwise identical mount, but with higher load carrying capacity. However they can also be used in other kinematic mount configurations, such as the four-legged mounts described below.
The implementation of Spacers is relatively straight forward, and is often used in context others than kinematic mounts. Conceptual cross-sections for some Spacer geometries are shown in FIG. 6 (All bodies shown are bodies of revolution), and embodiment of some Spacer implementations are shown in FIG. 7.
The simplest of these is a pair of spheres [6A] (“BiSphere”) that constrains the distance between two points on opposite sides of the mate. The BiSphere mating element clearly relies on single points of contact and so has a low load carrying capacity.
Other Spacer implementations include “Pill” [6B, 7A], “Lens” [6C, 7B], “Collar” [6D, 7C], and “Hat” [6E, 7D]. These are all able to carry higher loads compared to the BiSphere since they replace point contacts with CS interfaces by adding a mediating body that constrains the distance between two virtual centers (shown as center marks).
Each Spacer constrains this distance even as the two subassemblies move in relation to each other, but does not have to operate in all possible orientations—it suffices that it can operate as the two subassemblies move small distances around the nominal assembled position. The direction along which force is transferred in this nominal position is called the Spacer Axis of the Spacer and is shown as a dash-dot line in FIG. 6, and the constrained distance is labeled as “d”.
The angle between the Spacer Axis and the Mating Axis is called the “Tilt Angle” of the Spacer.
In most of the elements described above, there are movable mediating bodies that are introduced in order to eliminate point contacts and increase the load capacity of the mating element. These bodies also add their own DOFs to the system, but since their final position is not of significance to the kinematic mount, this does not require additional constraints in the system. Also, in practical implementations, the mediating bodies are loosely retained to one of the subassemblies of the mount so that they do not fall off when the subassemblies are moved around.
Loads acting on the assembly along the Mating Axis are called Axial loads, whereas loads acting in perpendicular directions are called Side loads. It is often the case that Axial loads are higher than Side loads, especially when the Mating Axis is vertical and the loads are primarily due to gravity.
In their physical embodiments, the mating surfaces of all these elements can be made from hardened metals such as steel or from hard ceramics or carbides, but can also be made from softer materials such as Aluminum or plastics.
In all kinematic mounts, there are six mechanical constraints that can be described by 6 equations, based on the mount's geometry. For the mount to function, the six equations must be independent. Taking only small deviations around the mount geometry, the six equations can be described as linear equations, and can thus be described by six vectors, forming a matrix. In theory, in order for the mount to function, the matrix must be invertible (non-singular). In practice, if the matrix is close to singular (known as “ill conditioned” matrices), the mount is highly non-rigid or unstable and so considered non-functional. This reasoning applies to the specification below, so when, for example, two direction vectors are specified to be “non-parallel” or “non-perpendicular”, the practical meaning is that even configurations where the vectors are close to being parallel or perpendicular (respectively) are disallowed. A minimum angle of 20° between vectors is generally enough to make sure they are sufficiently non-parallel, with 30° being preferable.